I NTERACTION OF I MPURITY ATOMS
IN B OSE -E INSTEIN C ONDENSATES
Alexander Klein and Michael Fleischhauer
Technische Universität Kaiserslautern, Germany
Abstract
Homogeneous condensate
The interaction of two spatially separated impurity atoms through phonon exchange in a BoseEinstein condensate is studied within a Bogoliubov
approach. The impurity atoms are held by deep
and narrow trap potentials and experience level
shifts which consist of a mean-field part and vacuum contributions from the Bogoliubov-phonons. In
addition there is a conditional energy shift resulting
from the exchange of phonons between the impurity atoms. Potential applications to quantum information processing are discussed.
Interacting, 1D trapped
condensate
Assume a condensate in a box with periodic boundary conditions, impurities located on the t -axis.
Energy shift
w
Condensate dynamics is solved within the Thomas-Fermi
approximation [1].
0 u
0 0
w y{z)|
Swj;Sx ^ €.‚ t 3ƒ
( }~ q %3/
* …„
MCv 0
0
h
w
w
w
‡† wZˆ w
ŒSŽ Ž u
with h
0 ‘ , the number of
x
4 S‰.Š =‹ , x
q
condensed atoms,x t) the width
of the impurity traps and %
The solutions of the Bogoliubov-de Gennes equation are
given by [2]
the distance between the impurities.
with
q *
:
«¬
(t /
®­
:Gº
:
«¬ ¹<
?
¬ · : 0
{¯ 4
S´ :
&Ž ¸
T
‚
t
t
T * ° ±'0 ² „¶µ
)°+±'²¥³
°+±'² 6
h
:
Y
The spatial dependence of the energy shift disappears and
one gets
Effective interaction of impurity
atoms in a BEC
0
q » M °±- ž².‰
0.1
∆
0
-ž
−0.1
Conditions
1
1
0
0
−0.2
tF- * t 0
¦ ¾ ¿
° ±'²½¼ I)À
€ÄÃ+Å St ° ±'²V¼‡Á¶Â
¿ ° '± ²Æ
0.5
0.4
Impurity atoms in sufficiently deep traps
state of traps occupied.
0.3
0.2
Only ground
0.1
1
‰ u
Here, ’
0 1 2
0
(z −z )/L
Interaction with condensate is given by
"!$#&% ')( %+*,%.
4 ! #&% ( %*5% 0
z0 /Lz
2
z
S“V”'• 0–S—.Mk0=v
.
•˜ ™ š — • –
Influence of the ratio
1
0
10
50
40
30
20
, where
•˜ ™ š •› •œ5
( %)/ ( %3/
0 1 2 1
( %3/ ( %3/ 6
/
/
with state dependent coupling constants 7 8
, -
1
with being the condensate wave function.
Furthermore, one is restricted to (coarse graining approximation)
J.K B
A - J.K B
A - - 4
0
* LFM 0
0
* LFM 0
0
* LFM 0
P 9
O A -(
A quantum phase gate can be implemented as follows
(2)
(3)
(
<
:
Energy shift
0
q *
:
”0 —
u
”0 4
ˆ
0
0
^€.
¡
* ¤t - ¥
V
* ¤t 0 ‹ ¤t ¡
( ¤t - / ¡ ( ¤t 0 /
)¡ £§
¦ ( t 0 4 t ”0 /
0
[Z
[ —C†
”0 4
0
0.5
7 MFp
A m no 7 7 M.p 4 7 T
T
T
T
-
-
4 )T T )T T M ( p - 4,q / Y
-
”
À.
Ô
s M.p
À TÔ
0
À 7 s
s
r
M.p
r
s
Ô 0 ¦
Ô ¦
-pulse
- ¦ -pulseÙ]
Ú 7 ÙÛ
Ú 7 - -pulseÙ
7)7 T
T
ÙÜ À 7 Ù
À 7 Ù * À 7 *
*
Ù Ú À.
Ù Ú À.
Ù
7 À.
* À3À.
ÙÝ À3T À.
ÙÞ À3T À.
Ùß
and
7 are the logical states.
Gate time has to be long compared to the inverse condiq _ -.
tional frequency shift
Because of (4) the major restriction is the trap frequency in
p ”
weak (longitudinal) direction .
zB = 10 z0
zB = 20 z0
zB = 50 z0
z = 100 z
B
7)7 À 7 7 Àk
À3Àk
M q
r
0
0
7)7 −0.4
rs
rs
M p
-
s
References
−0.5
-
M q r
∆
M p
7 T
T)T
- M ž Ÿ Mk£kT p
”
¡3¢ -
t (t
t / , ¤t
t
t
t , t the width
where ¤t
of the condensate
— t)¦© ¨ the
trap,
0ª width of the impurity traps.
(
Furthermore - ž
/ with the radial confinement
¨ 0ª "M&— “ ” p ª . ¡ are Hermite polynomials.
( %)/*,? ( %3/ /
:
; /*,?&@ ( % ; / / Y
Eqs. (1)-(3) correspond to effective coarse grained Hamiltonian for (symmetric located) impurity atoms in interaction
picture. It reads
7 T
.À T Ô
Ideal, 1D trapped condensate
’s are the Bogoliubov energies and
[ 0 1
( g g ; / ! #&% '3( %*5% /
3( %3/
:
0
i !j#k% ;Gl '3( % ; *,% [ \ / l 1 F( % ; / ( < @ ( %
l
l
×T)Ø ÕÖT
Ô 0 ×××× ÕÕÕÕ Ô
ÕÕ ×× ×
ÕÕ
×
× ×Ø
7 ÕÖ Õ Õ Õ
× × 7 T T
Ô ÕÕÕÕ ×××× Ô 0
Õ×
7)7 1D geometry preferred.
7 À. s
K K \e f : P+[ 9 P+[ \ 9 ]3:8;=^)_a` bCc d _
(g g;/
O A ( / A ( ; /Q
:
f :
7SÏSÐÑ
(1)
Correlation functions are given by
where the h
i
(4)
Implementation of a quantum
phase gate
K
9
9 P 9 P 9
! K N)# ; B A - 3( ; /'O A - ( / A - ( ; /Q
K
9
9 P 9 P 9
! K N)# ; B A - 3( ; /'O A 0 ( / A 0 ( ; /Q
K
9
9
P 9 P 9
! K N)# ; B A - - 3( ; /SRO A - ( / A - ( ; /Q
P 9 4
XWZY
terms with TVU
/ A 0 ( ; /Q
u 0
p ” ÌË - ž
Y
M 0 °+'±0 ² p ”‘Î
T{Í
p ª Reduced
operator matrix elements B A C DFE
G 9) B HI3statistical
9
A
for the two impurity atoms are calculated in
the usual Born approximation
J.K B
A - Á¶È ɶÊ
¦ T
For a tight transverse confinement
, a scat 737
tering length ¨
ÉkÁ between impurities and conden"Ò
¨
sate and
one finds a condiÉkÁ within the condensate
i T 73Ó=ÐÑ .
tional frequency shift of ¦
.
:
:
:
:
9 1
>
> 2
3:8;=<
3( %3/ 4
( %)/ ,
/
* ?&@ ( %3/
(%
with
the Thomas-Fermi radius,
the distance of
the impurities from the edge of the condensate, and ¿
M.p ” — )´
.
q®Ç
Dynamics of the condensate is calculated by standard Bogoliubov approach [1]
1
I)À
°±²
K
−0.6
−0.5
[1] P. Öhberg et al., “Low-energy elementary excitations of a
trapped Bose-condensed gas”, Phys. Rev. A, 56, R3346
(1997).
−0.7
−0.8
−0.9
−1
0
−1
0
In this figure ¤t -
0.5
*V¤t 0
Y
1
¤t -
0.05
1.5
0.1
0.15
2
2.5
Influence of the width of the impurity traps is very weak.
[2] D. S. Petrov, G. V. Shlyapnikov, J. T. M. Walraven,
“Regimes of quantum degeneracy in trapped 1D gases”,
Phys. Rev. Lett., 85, 3745 (2000).
[3] A. Klein, M. Fleischhauer, “Interaction of impurity atoms
in Bose-Einstein condensates”, cond-mat/0407809.